Further, the functor tensor product is a left Quillen bifunctor when endowing the two source categories with any of the pairs of model structures (projective, injective), (Reedy, Reedy), or (injective, projective) [1]}.
Further, the functor \(N({\mathbbe {\Delta }}_{/(-)})^\mathrm {op}\colon {\mathbbe {\Delta }}\longrightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  is projectively cofibrant [2]}.
(Note that it is then also Reedy cofibrant.)
Consequently, the functor
\(N({\mathbbe {\Delta }}_{/(-)})^\mathrm {op}\underset{{\mathbbe {\Delta }}^\mathrm {op}}{\otimes } (-)\colon ({{K}}_\infty ^p)^{{\mathbbe {\Delta }}^\mathrm {op}} \longrightarrow {{K}}_\infty ^p\) 
